The following is a report containing the models fit to data from 2019-11-29 to 2021-01-01. Data are measured in XXX Millions. This report summarizes results for Net Foreign Assets.

RMSE

The RMSE is defined as

\[RMSE=\sqrt{\sum_{t\in\mathcal{T_{eval}}}(y_t-\hat{y}_t(h))^2}\] where \(y_t\) is the observed value and \(\hat{y}_t(h)\) is the h-step ahead forecast of \(y_t\). The RMSE for each model and HORIZON is shown below. Lower values indicate better forecasts, the best individual model is in bold, while combinations are colored.

MAE

The MAE is defined as

\[MAE=\sum_{t\in\mathcal{T_{eval}}} \left| y_t-\hat{y}_t(h) \right|\] where \(y_t\) is the observed value and \(\hat{y}_t(h)\) is the h-step ahead forecast of \(y_t\). The RMSE for each model and HORIZON is shown below. Lower values indicate better forecasts.

Hit Ratio

To evaluate the accuracy of a 95% prediction interval, one can consider the proportion of times the observed value is outside the prediction interval. This is summarised in the table below. Values close to 0.05 indicate the best forecasts with values below 0.05 indicating conservative prediction intervals.

Net Foreign Assets

Net Foreign Assets over the training period is plotted below. Missing values (mostly Fridays and Saturdays) are linearly interpolated.

While the mean of Net Foreign Assets follows a random walk, the first difference does exhibit conditional heteroskedasticity.

Conditional variances

Since the fitted values of a GARCH model are simple equal to the (constant) sample mean, the data are plotted with 95% intervals. This shows how each model captures time varying volatility.

Model Descriptions and Parameters

EWMA

A exponentially weighted moving average (EWMA) model is a special case of the GARCH model with a single parameter. The estimated parameters are summarised below:

——————————— * GARCH Model Fit ———————————*

Conditional Variance Dynamics

GARCH Model : iGARCH(1,1) Mean Model : ARFIMA(0,0,0) Distribution : norm

Optimal Parameters

    Estimate  Std. Error   t value Pr(>|t|)

mu -0.000672 0.014425 -0.046621 0.96281 omega 0.000000 NA NA NA alpha1 0.032530 0.003880 8.383309 0.00000 beta1 0.967470 NA NA NA

Robust Standard Errors: Estimate Std. Error t value Pr(>|t|) mu -0.000672 0.018183 -0.036984 0.970498 omega 0.000000 NA NA NA alpha1 0.032530 0.010252 3.172936 0.001509 beta1 0.967470 NA NA NA

LogLikelihood : -5004.499

Information Criteria

Akaike 2.7425 Bayes 2.7459 Shibata 2.7425 Hannan-Quinn 2.7437

Weighted Ljung-Box Test on Standardized Residuals

                    statistic p-value

Lag[1] 416.6 0 Lag[2*(p+q)+(p+q)-1][2] 419.6 0 Lag[4*(p+q)+(p+q)-1][5] 438.7 0 d.o.f=0 H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals

                    statistic p-value

Lag[1] 208.0 0 Lag[2*(p+q)+(p+q)-1][5] 218.7 0 Lag[4*(p+q)+(p+q)-1][9] 239.7 0 d.o.f=2

Weighted ARCH LM Tests

        Statistic Shape Scale   P-Value

ARCH Lag[3] 7.765 0.500 2.000 5.327e-03 ARCH Lag[5] 20.116 1.440 1.667 2.003e-05 ARCH Lag[7] 39.415 2.315 1.543 5.090e-10

Nyblom stability test

Joint Statistic: 0.4042 Individual Statistics:
mu 0.004732 alpha1 0.402164

Asymptotic Critical Values (10% 5% 1%) Joint Statistic: 0.61 0.749 1.07 Individual Statistic: 0.35 0.47 0.75

Sign Bias Test

               t-value      prob sig

Sign Bias 1.726 8.436e-02 Negative Sign Bias 6.101 1.167e-09 Positive Sign Bias 10.777 1.114e-26 Joint Effect 153.722 4.147e-33 **

Adjusted Pearson Goodness-of-Fit Test:

group statistic p-value(g-1) 1 20 574.1 1.489e-109 2 30 636.8 1.491e-115 3 40 712.2 1.342e-124 4 50 726.0 2.696e-121

Elapsed time : 0.05102801

GARCH

An standard GARCH(1,1) is regularly used to model conditional heteroskedasticty. The estimated parameters are summarised below:

——————————— * GARCH Model Fit ———————————*

Conditional Variance Dynamics

GARCH Model : sGARCH(1,1) Mean Model : ARFIMA(0,0,0) Distribution : norm

Optimal Parameters

    Estimate  Std. Error  t value Pr(>|t|)

mu 0.001849 0.012821 0.14421 0.88533 omega 0.571130 0.026051 21.92335 0.00000 alpha1 0.488887 0.039417 12.40285 0.00000 beta1 0.013750 0.021109 0.65136 0.51481

Robust Standard Errors: Estimate Std. Error t value Pr(>|t|) mu 0.001849 0.009785 0.18895 0.85013 omega 0.571130 0.058849 9.70496 0.00000 alpha1 0.488887 0.073568 6.64536 0.00000 beta1 0.013750 0.040633 0.33840 0.73507

LogLikelihood : -4875.447

Information Criteria

Akaike 2.6729 Bayes 2.6797 Shibata 2.6729 Hannan-Quinn 2.6754

Weighted Ljung-Box Test on Standardized Residuals

                    statistic p-value

Lag[1] 159.2 0 Lag[2*(p+q)+(p+q)-1][2] 170.3 0 Lag[4*(p+q)+(p+q)-1][5] 200.4 0 d.o.f=0 H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals

                    statistic   p-value

Lag[1] 0.3028 5.821e-01 Lag[2*(p+q)+(p+q)-1][5] 1.2214 8.079e-01 Lag[4*(p+q)+(p+q)-1][9] 21.6888 7.334e-05 d.o.f=2

Weighted ARCH LM Tests

        Statistic Shape Scale   P-Value

ARCH Lag[3] 0.06307 0.500 2.000 8.017e-01 ARCH Lag[5] 1.42629 1.440 1.667 6.119e-01 ARCH Lag[7] 26.27941 2.315 1.543 1.526e-06

Nyblom stability test

Joint Statistic: 5.1086 Individual Statistics:
mu 0.09198 omega 3.79540 alpha1 0.54163 beta1 1.14985

Asymptotic Critical Values (10% 5% 1%) Joint Statistic: 1.07 1.24 1.6 Individual Statistic: 0.35 0.47 0.75

Sign Bias Test

               t-value   prob sig

Sign Bias 0.04015 0.9680
Negative Sign Bias 0.95582 0.3392
Positive Sign Bias 0.30883 0.7575
Joint Effect 1.67421 0.6427

Adjusted Pearson Goodness-of-Fit Test:

group statistic p-value(g-1) 1 20 510.0 4.518e-96 2 30 549.9 1.525e-97 3 40 554.0 3.041e-92 4 50 578.7 1.306e-91

Elapsed time : 0.2722399

eGARCH

An eGARCH(1,1) extends the GARCH model by allowings for asymmetric effects. The estimated parameters are summarised below:

——————————— * GARCH Model Fit ———————————*

Conditional Variance Dynamics

GARCH Model : eGARCH(1,1) Mean Model : ARFIMA(0,0,0) Distribution : norm

Optimal Parameters

    Estimate  Std. Error  t value Pr(>|t|)

mu 0.012058 0.013732 0.87808 0.379901 omega -0.041826 0.024244 -1.72523 0.084486 alpha1 0.052851 0.024117 2.19144 0.028420 beta1 0.288775 0.045214 6.38683 0.000000 gamma1 0.655781 0.034563 18.97359 0.000000

Robust Standard Errors: Estimate Std. Error t value Pr(>|t|) mu 0.012058 0.013780 0.87504 0.381552 omega -0.041826 0.052070 -0.80327 0.421821 alpha1 0.052851 0.035173 1.50261 0.132939 beta1 0.288775 0.087746 3.29102 0.000998 gamma1 0.655781 0.053105 12.34883 0.000000

LogLikelihood : -4862.87

Information Criteria

Akaike 2.6666 Bayes 2.6751 Shibata 2.6666 Hannan-Quinn 2.6696

Weighted Ljung-Box Test on Standardized Residuals

                    statistic p-value

Lag[1] 162.8 0 Lag[2*(p+q)+(p+q)-1][2] 170.2 0 Lag[4*(p+q)+(p+q)-1][5] 198.3 0 d.o.f=0 H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals

                    statistic   p-value

Lag[1] 0.532 4.658e-01 Lag[2*(p+q)+(p+q)-1][5] 1.847 6.551e-01 Lag[4*(p+q)+(p+q)-1][9] 23.291 2.786e-05 d.o.f=2

Weighted ARCH LM Tests

        Statistic Shape Scale   P-Value

ARCH Lag[3] 1.243 0.500 2.000 2.650e-01 ARCH Lag[5] 2.408 1.440 1.667 3.880e-01 ARCH Lag[7] 28.515 2.315 1.543 3.966e-07

Nyblom stability test

Joint Statistic: 5.2232 Individual Statistics:
mu 0.08241 omega 3.74179 alpha1 0.21411 beta1 2.60814 gamma1 2.83494

Asymptotic Critical Values (10% 5% 1%) Joint Statistic: 1.28 1.47 1.88 Individual Statistic: 0.35 0.47 0.75

Sign Bias Test

               t-value   prob sig

Sign Bias 0.2011 0.8407
Negative Sign Bias 0.1403 0.8884
Positive Sign Bias 0.3311 0.7406
Joint Effect 0.2891 0.9621

Adjusted Pearson Goodness-of-Fit Test:

group statistic p-value(g-1) 1 20 530.0 2.833e-100 2 30 555.5 1.052e-98 3 40 578.7 2.890e-97 4 50 637.1 2.589e-103

Elapsed time : 0.4344161

gjrGARCH

A GJR GARCH(1,1) extends the GARCH model by allowings for asymmetric effects. The estimated parameters are summarised below:

——————————— * GARCH Model Fit ———————————*

Conditional Variance Dynamics

GARCH Model : gjrGARCH(1,1) Mean Model : ARFIMA(0,0,0) Distribution : norm

Optimal Parameters

    Estimate  Std. Error  t value Pr(>|t|)

mu 0.018459 0.014204 1.29949 0.193775 omega 0.574773 0.027025 21.26814 0.000000 alpha1 0.585565 0.059663 9.81446 0.000000 beta1 0.014462 0.022947 0.63023 0.528541 gamma1 -0.209572 0.075691 -2.76879 0.005626

Robust Standard Errors: Estimate Std. Error t value Pr(>|t|) mu 0.018459 0.011244 1.64171 0.100651 omega 0.574773 0.057909 9.92540 0.000000 alpha1 0.585565 0.112390 5.21012 0.000000 beta1 0.014462 0.041721 0.34664 0.728864 gamma1 -0.209572 0.123409 -1.69819 0.089472

LogLikelihood : -4871.608

Information Criteria

Akaike 2.6714 Bayes 2.6799 Shibata 2.6714 Hannan-Quinn 2.6744

Weighted Ljung-Box Test on Standardized Residuals

                    statistic p-value

Lag[1] 160.3 0 Lag[2*(p+q)+(p+q)-1][2] 171.0 0 Lag[4*(p+q)+(p+q)-1][5] 200.8 0 d.o.f=0 H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals

                    statistic   p-value

Lag[1] 0.3346 0.5629747 Lag[2*(p+q)+(p+q)-1][5] 1.2607 0.7984087 Lag[4*(p+q)+(p+q)-1][9] 21.0558 0.0001072 d.o.f=2

Weighted ARCH LM Tests

        Statistic Shape Scale   P-Value

ARCH Lag[3] 0.00388 0.500 2.000 9.503e-01 ARCH Lag[5] 1.56501 1.440 1.667 5.754e-01 ARCH Lag[7] 25.57290 2.315 1.543 2.332e-06

Nyblom stability test

Joint Statistic: 5.1464 Individual Statistics:
mu 0.08485 omega 3.83598 alpha1 0.54184 beta1 1.39182 gamma1 0.32691

Asymptotic Critical Values (10% 5% 1%) Joint Statistic: 1.28 1.47 1.88 Individual Statistic: 0.35 0.47 0.75

Sign Bias Test

               t-value   prob sig

Sign Bias 0.2526 0.8006
Negative Sign Bias 0.2376 0.8122
Positive Sign Bias 0.3511 0.7255
Joint Effect 0.2419 0.9706

Adjusted Pearson Goodness-of-Fit Test:

group statistic p-value(g-1) 1 20 514.2 5.941e-97 2 30 532.8 5.012e-94 3 40 570.2 1.512e-95 4 50 604.6 8.826e-97

Elapsed time : 0.5956001